Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 29 / 1 / i

A causal time-series representation uses only the present and past driving white noise. Thus the coefficient condition isThe series must have its stated convergence meaning. For centered white noise of positive finite variance, ${\sum }_{r\ge 0}|a_r{|}^2<\infty$ is sufficient and necessary for mean-square convergence. In the usual stable-filter convention one imposes the stronger ${\sum }_{r\ge 0}|a_r|<\infty$. A bilateral stationary linear process need not be causal: terms with $r<0$ involve future driving values.